Bird Flight V Formation as Geometry Problem of Max Visual Contact

By Xah Lee. Date: 2010-03-05

Why The V Formation?

Was reading Wikipedia V formation, that is, the formation of flight of geese, ducks. The interesting thing is, why they fly in that formation. The Wikipedia article is quite interesting, but too short.

The reason given by Wikipedia are basically two: ① for the whole group, the V formation is more efficient for (energy cost)/(distance traveled), and increases flight distance up to 71%. ② Visual contact is part of the reason for V formation.

It is interesting to note that the birds in V Formation rotates. The bird in front is the most tiring. Though, Wikipedia doesn't say how often they rotate, or how they rotate, or by what biological behavior means they signal a rotation, i.e. instinctively and regularly by bio-clock? or whenever the lead bird feel tired? Does he simply just slow down? Or does he do some kinda dance? And, where do he drop off to, the all the way to the bottom of the rung or just second from lead?

Math Problem Of Maximizing Visual Contact

I thought a bit about the visual contact reason. It is essentially
a mathematical problem. Each bird has max visual contact in front or
to the sides. You have a group of birds. The problem is to find a
arrangement so that the visual contact between the birds as a group is
maximized. Because the eyes of birds are essentially on the sides, not on top of their heads,
so this pretty much means the best arrangement must happen in a
horizontal plane. Clearly, the max visual contact can't
be the only reason for V formation, because a half circle formation is
better. In a V formation, half of it is linear, so the bird in front
of you blocks your view of the whole line. Also, i thought about
inverted V formation. I think inverted V might be a better formation
with respect to maximizing the whole group's visual contact.

To be a bit precise, let's assume there are 5 birds. A goose's
field of vision
is 300° forward (a random guess). Human's field of vision is perhaps 160°. To compute the max visual contact, we may weight each bird's visual contact of another bird by the degree the bird is in focus. For example, if person A stands in front of you, and person B stand on your left barely visible, sure both are in visual contact, but you'd give a much higher score for your visual contact with A than B. For the 5 birds, let's name tham A B C D E. For bird A, we compute the weighted score for visual contact of A with B, then A with C, etc. So we have 4 numbers. Do this for each bird. So in the end we have 4*5 = 20 numbers. Average them would give as the visual contact score of the whole group.

Instead of solving the max visual contact problem for birds, let's
change the problem to humans, which makes it more interesting to
us. And, let's just simplify the problem for people standing on the
ground. So, given n person, what's the best arrangement for max visual
contact of the whole group? (assume we cannot turn our head)

Clearly, when the problem is put in this way, one immediately see
that the size of the group matters, and the distance between person
also matters. For example, if you have 1000 person, the best
arrangement may not be some V formation. A more dense, grid formation
may do much better. This is probably because we haven't taken into
account of distance in visual contact. If a person is 1 km away, you
may not consider it to be in visual contact.

So, at this point, we might want to change our problem by limit the
number to say 100, then we don't have to worry about the visual distance much.

However, the math problem without the size constraint or visual distance complexity is still interesting one. Here's another try at the pure math formulation:
Given n points in a plane, assume each point has forward visual field of α°, and assume point A has better visual contact of point B if B is more directly ahead of A, what is the best arrangement of the points so that visual contact is maximized for the whole group?

To give the problem a slight more practical touch, we may consider
the dots to be circles of radius 1, so that we introduce into the
problem of the blocking of sight issue. This change basically means
whatever results will be scaled up so position between birds can be
say 1000. The larger, the better. This makes the problem less
interesting, so we'll introduce another constraint, that the position
between 2 most distant birds cannot be more than say n*100. Also, a full view of a
circle is better than half view. So, we'll have to add the weight of
the visual contact so that, a bird directly in front of you is better
than a bird to the side, but also, a full view of a bird is better
than half view. The exact weighted score formula for visual contact
needs to be worked out a bit.